The boundedness of classical operators on variable L-p spaces

Annales Academiæ Scien ia um Fennicæ
Ma hema ica
Volumen 31, 2006, 239–264
THE BOUNDEDNESS OF CLASSICAL OPERATORS
ON VARIABLE LpSPACES
D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
T ini y College, Depa men o Ma hema ics
Ha o d, CT 06106-3100, U.S.A.; da id.c uzu ib[email p o ec ed]
Uni e si ´a di Napoli, Dipa imen o di Cos uzioni e Me odi Ma ema ici in A chi e u a
Via Mon eoli e o, 3, IT-80134 Napoli, I aly and
Consiglio Nazionale delle Rice che, Is i u o pe le Applicazioni del Calcolo “Mau o Picone”
Sezione di Napoli, ia Pie o Cas ellino, 111, IT-80131 Napoli, I aly; [email p o ec ed]
Uni e sidad Au ´onoma de Mad id, Depa amen o de Ma em´a icas
ES-28049 Mad id, Spain; [email p o ec ed]
Uni e sidad de Se illa, Depa amen o de An´alisis Ma em´a ico
Facul ad de Ma em´a icas, ES-41080 Se illa, Spain; ca losp[email p o ec ed]
Abs ac . We show ha many classical ope a o s in ha monic analysis—such as maximal
ope a o s, singula in eg als, commu a o s and ac ional in eg als—a e bounded on he a iable
Lebesgue space Lp(·)whene e he Ha dy–Li lewood maximal ope a o is bounded on Lp(·).
Fu he , we show ha such ope a o s sa is y ec o - alued inequali ies. We do so by applying he
heo y o weigh ed no m inequali ies and ex apola ion.
As applica ions we p o e he Calde ´on–Zygmund inequali y o solu ions o 4u= in
a iable Lebesgue spaces, and p o e he Calde ´on ex ension heo em o a iable Sobole spaces.
1. In oduc ion
Gi en an open se Ω ⊂Rn, we conside a measu able unc ion p: Ω −→
[1,∞), Lp(·)(Ω) deno es he se o measu able unc ions on Ω such ha o
some λ > 0, ZΩ| (x)|
λp(x)
dx < ∞.
This se becomes a Banach unc ion space when equipped wi h he no m
k kp(·),Ω= in λ > 0 : ZΩ| (x)|
λp(x)
dx ≤1.
These spaces a e e e ed o as a iable Lebesgue spaces o , mo e simply, as a i-
able Lpspaces, since hey gene alize he s anda d Lpspaces: i p(x) = p0is
2000 Ma hema ics Subjec Classi ica ion: P ima y 42B25, 42B20, 42B15, 35J05.
The hi d au ho is pa ially suppo ed by MEC G an MTM2004-00678, and he ou h
au ho is pa ially suppo ed by DGICYT G an PB980106.
240 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
cons an , hen Lp(·)(Ω) equals Lp0(Ω). (He e and below we w i e p(·) ins ead
o p o emphasize ha he exponen is a unc ion and no a cons an .) They ha e
many p ope ies in common wi h he s anda d Lpspaces.
These spaces, and he co esponding a iable Sobole spaces Wk,p(·)(Ω), a e
o in e es in hei own igh , and also ha e applica ions o pa ial di e en ial
equa ions and he calculus o a ia ions. (See, o example, [1], [12], [15], [19],
[30], [39], [46] and hei e e ences.)
In many applica ions, a c ucial s ep has been o show ha one o he clas-
sical ope a o s o ha monic analysis—e.g., maximal ope a o s, singula in eg als,
ac ional in eg als—is bounded on a a iable Lpspace. Many au ho s ha e con-
side ed he ques ion o su icien condi ions on he exponen unc ion p(·) o
gi en ope a o s o be bounded: see, o example, [13], [15], [27], [28], [29], [40].
Ou app oach is di e en . Ra he han conside es ima es o indi idual op-
e a o s, we apply echniques om he heo y o weigh ed no m inequali ies and
ex apola ion o show ha he boundedness o a wide a ie y o ope a o s ollows
om he boundedness o he maximal ope a o on a iable Lpspaces, and om
known es ima es on weigh ed Lebesgue spaces. In o de o p o ide he ounda ion
o s a ing ou esul s, we discuss each o hese ideas in u n.
The maximal ope a o . In ha monic analysis, a undamen al ope a o
is he Ha dy–Li lewood maximal ope a o . Gi en a unc ion , we de ine he
maximal unc ion, M , by
M (x) = sup
Q3x
1
|Q|ZQ
| (y)|dy,
whe e he sup emum is aken o e all cubes con aining x. I is well known ha
Mis bounded on Lp, 1 <p<∞, and i is na u al o ask o which exponen
unc ions p(·) he maximal ope a o is bounded on Lp(·)(Ω). Fo conciseness,
de ine P(Ω) o be he se o measu able unc ions p: Ω −→ [1,∞) such ha
p−= ess in {p(x) : x∈Ω}>1, p+= ess sup{p(x) : x∈Ω}<∞.
Le B(Ω) be he se o p(·)∈P(Ω) such ha Mis bounded on Lp(·)(Ω).
Theo em 1.1. Gi en an open se Ω⊂Rn, and p(·)∈P(Ω), suppose ha
p(·)sa is ies
(1.1) |p(x)−p(y)| ≤ C
−log(|x−y|), x, y ∈Ω,|x−y| ≤ 1/2,
(1.2) |p(x)−p(y)| ≤ C
log(e+|x|), x, y ∈Ω,|y| ≥ |x|.
Then p(·)∈B(Ω), ha is, he Ha dy–Li lewood maximal ope a o is bounded
on Lp(·)(Ω).
The boundedness o classical ope a o s on a iable Lpspaces 241
Theo em 1.1 is independen ly due o C uz-U ibe, Fio enza and Neugebaue
[10] and o Nek inda [35]. (In ac , Nek inda eplaced (1.2) wi h a sligh ly mo e
gene al condi ion.) Ea lie , Diening [12] showed ha (1.1) alone is su icien i
Ω is bounded. Examples show ha he con inui y condi ions (1.1) and (1.2) a e
in some sense close o necessa y: see Pick and R˚uˇziˇcka [37] and [10]. See also
he examples in [33]. The condi ion p−>1 is necessa y o M o be bounded;
see [10].
Ve y ecen ly, Diening [14], wo king in he mo e gene al se ing o Musielak–
O licz spaces, has gi en a necessa y and su icien condi ion on p(·) o M o be
bounded on Lp(·)(Rn). His exac condi ion is somewha echnical and we e e
he eade o [14] o de ails.
Because ou p oo s ely on duali y a gumen s, we will no need ha he
maximal ope a o is bounded on Lp(·)(Ω) bu on i s associa e space Lp0(·)(Ω),
whe e p0(·) is he conjuga e exponen unc ion de ined by
1
p(x)+1
p0(x)= 1, x ∈Ω.
Since
|p0(x)−p0(y)| ≤ |p(x)−p(y)|
(p−−1)2,
i ollows a once ha i p(·) sa is ies (1.1) and (1.2), hen so does p0(·)—i.e.,
i hese wo condi ions hold, hen Mis bounded on Lp(·)(Ω) and Lp0(·)(Ω).
Fu he mo e, Diening’s cha ac e iza ion o a iable Lpspaces on which he max-
imal ope a o is bounded has he ollowing impo an consequence (see [14, The-
o em 8.1]).
Theo em 1.2. Le p(·)∈P(Rn). Then he ollowing condi ions a e equi -
alen :
(a) p(·)∈B(Rn).
(b) p0(·)∈B(Rn)
(c) p(·)/q ∈B(Rn) o some 1< q < p−.
(d) p(·)/q0∈B(Rn) o some 1< q < p−.
Weigh s and ex apola ion. By a weigh we mean a non-nega i e, locally
in eg able unc ion w. The e is a as li e a u e on weigh s and weigh ed no m
inequali ies; he e we will summa ize he mos impo an aspec s, and we e e he
eade o [17], [21] and hei e e ences o comple e in o ma ion.
Cen al o he s udy o weigh s a e he so-called Apweigh s, 1 ≤p≤ ∞.
When 1 <p<∞, we say w∈Api o e e y cube Q,
1
|Q|ZQ
w(x)dx 1
|Q|ZQ
w(x)1−p0dxp−1
≤C < ∞.
242 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
We say ha w∈A1i Mw(x)≤Cw(x) o a.e. x. I 1 ≤p < q < ∞, hen
Ap⊂Aq. We le A∞deno e he union o all he Apclasses, 1 ≤p < ∞.
Weigh ed no m inequali ies a e gene ally o wo ypes. The i s is
(1.3) ZRn
|T (x)|p0w(x)dx ≤CZRn
| (x)|p0w(x)dx,
whe e Tis some ope a o and w∈Ap0, 1 < p0<∞. (In o he wo ds, Tis
de ined and bounded on Lp0(w).) The cons an is assumed o depend only on he
Ap0cons an o w. The second ype is
(1.4) ZRn
|T (x)|p0w(x)dx ≤CZRn
|S (x)|p0w(x)dx,
whe e Sand Ta e ope a o s, 0 < p0<∞,w∈A∞, and is such ha he le -
hand side is ini e. The cons an is assumed o depend only on he A∞cons an
o w. Such inequali ies a e known o a wide a ie y o ope a o s and pai s o
ope a o s. (See [17], [21].)
Co esponding o hese ypes o inequali ies a e wo ex apola ion heo ems.
Associa ed wi h (1.3) is he classical ex apola ion heo em o Rubio de F ancia [38]
(also see [17], [21]). He p o ed ha i (1.3) holds o some ope a o T, a ixed alue
p0, 1 < p0<∞, and e e y weigh w∈Ap0, hen (1.3) holds wi h p0 eplaced
by any p, 1 < p < ∞, whene e w∈Ap. Recen ly, he analogous ex apola ion
esul o inequali ies o he o m (1.4) was p o ed in [11]: i (1.4) holds o some
p0, 0 < p0<∞and e e y w∈A∞, hen i holds o e e y p, 0 <p<∞. (Mo e
gene al e sions o hese esul s will be s a ed in Sec ion 6 below.)
1.1. Main esul s. The p oo s o he abo e ex apola ion heo ems depend
no on he p ope ies o he ope a o s, bu a he on duali y, he s uc u e o
Apweigh s, and no m inequali ies o he Ha dy–Li lewood maximal ope a o .
These ideas can be ex ended o he se ing o a iable Lpspaces o yield ou
main esul , which can be summa ized as ollows: I an ope a o T, o a pai
o ope a o s (T, S), sa is ies weigh ed no m inequali ies on he classical Lebesgue
spaces, hen i sa is ies he co esponding inequali y in a a iable Lpspace on
which he maximal ope a o is bounded.
To s a e and p o e ou main esul , we will adop he app oach aken in [11].
The e i was obse ed ha since no hing is assumed abou he ope a o s in ol ed
(e.g., linea i y o sublinea i y), i is be e o eplace inequali ies (1.3) and (1.4)
wi h
(1.5) ZRn
(x)p0w(x)dx ≤CZRn
g(x)p0w(x)dx,
whe e he pai s ( , g) a e such ha he le -hand side o he inequali y is ini e.
One impo an consequence o adop ing his app oach is ha ec o - alued in-
equali ies ollow immedia ely om ex apola ion.
The boundedness o classical ope a o s on a iable Lpspaces 243
He ea e Fwill deno e a amily o o de ed pai s o non-nega i e, measu able
unc ions ( , g). Whene e we say ha an inequali y such as (1.5) holds o any
( , g)∈Fand w∈Aq( o some q, 1 ≤q≤ ∞), we mean ha i holds o any
pai in Fsuch ha he le -hand side is ini e, and he cons an Cdepends only
on p0and he Aqcons an o w.
Finally, no e ha in he classical Lebesgue spaces we can wo k wi h Lpwhe e
0< p < 1. (Thus, in (1.4) o (1.5) we can ake p0<1.) We would like o
conside analogous spaces wi h a iable exponen s. De ine P0(Ω) o be he se
o measu able unc ions p: Ω −→ (0,∞) such ha
p−= ess in {p(x) : x∈Ω}>0, p+= ess sup{p(x) : x∈Ω}<∞.
Gi en p(·)∈P0(Ω), we can de ine he space Lp(·)(Ω) as abo e. This is equi -
alen o de ining i o be he se o all unc ions such ha | |p0∈Lq(·)(Ω),
whe e 0 < p0< p−and q(x) = p(x)/p0∈P(Ω). We can de ine a quasi-no m on
his space by
k kp(·),Ω=
| |p0
1/p0
q(·),Ω.
We will no need any o he p ope ies o hese spaces, so his de ini ion will su ice
o ou pu poses.
Theo em 1.3. Gi en a amily Fand an open se Ω⊂Rn, suppose ha
o some p0,0< p0<∞, and o e e y weigh w∈A1,
(1.6) ZΩ
(x)p0w(x)dx ≤C0ZΩ
g(x)p0w(x)dx, ( , g)∈F,
whe e C0depends only on p0and he A1cons an o w. Le p(·)∈P0(Ω) be
such ha p0< p−, and p(·)/p00∈B(Ω). Then o all ( , g)∈Fsuch ha
∈Lp(·)(Ω),
(1.7) k kp(·),Ω≤Ckgkp(·),Ω,
whe e he cons an Cis independen o he pai ( , g).
We wan o call a en ion o wo ea u es o Theo em 1.3. Fi s , he conclusion
(1.7) is an a p io i es ima e: ha is, i holds o all ( , g)∈Fsuch ha ∈
Lp(·)(Ω). In p ac ice, when applying his heo em in conjunc ion wi h inequali ies
o he o m (1.3) o show ha an ope a o is bounded on a iable Lpwe will
usually need o wo k wi h a collec ion o unc ions which sa is y he gi en
weigh ed Lebesgue space inequali y and a e dense in Lp(·)(Ω). When wo king
wi h inequali ies o he o m (1.3) he inal es ima e will hold o a sui able amily
o “nice” unc ions.
Second, he amily Fin he hypo hesis o and conclusion o Theo em 1.7 is
he same, so he goal is o ind a la ge, easonable amily Fsuch ha (1.6) holds
wi h a cons an depending only on p0and he A1cons an o w.

244 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
Rema k 1.4. In Theo em 1.3, (1.7) holds i p(·) sa is ies (1.1) and (1.2).
By Theo em 1.1, se ing q(x) = p(x)/p0we ha e ha q(·)∈P(Ω) and
|q0(x)−q0(y)| ≤ |p(x)−p(y)|
p0(p−/p0−1)2.
Rema k 1.5. When Ω = Rn, i 1 ≤p0< p−, hen by Theo em 1.2 he hy-
po hesis ha p(·)/p00∈B(Rn) is equi alen o assuming ha p(·)∈B(Rn).
As we will see below, his will allow us o conclude ha a a ie y o ope a o s a e
bounded on Lp(·)(Rn) whene e he Ha dy–Li lewood maximal ope a o is.
Rema k 1.6. Ou app oach using pai s o unc ions leads o an equi alen
o mula ion o Theo em 1.3 in which he exponen p0does no play a ole. This
can be done by de ining a new amily Fp0consis ing o he pai s ( p0, gp0) wi h
( , g)∈F. No ice ha in his case (1.6) is sa is ied by Fp0wi h p0= 1. Thus,
he case p0= 1 will imply ha i 1 < p−and p(·)0∈B(Ω) hen (1.7) holds.
The e o e, i we de ine (x) = p(x)p0, we ha e ha (·)∈P0(Ω), p0< −,
( (·)/p0)0∈B(Ω) and (1.7) holds wi h (·) in place o p(·). Bu his is exac ly
he conclusion o Theo em 1.3.
Rema k 1.7. We belie e ha a mo e gene al e sion o Theo em 1.3 is ue,
one which holds o la ge classes o weigh s and yields inequali ies in weigh ed
a iable Lpspaces. Howe e , p o ing such a esul will equi e a weigh ed e sion
o Theo em 1.1, and e en he s a emen o such a esul has eluded us. Fo such
a weigh ed ex apola ion esul he app op ia e class o weigh s is no longe A1,
bu Ap(as in [38]) o A∞(as in [11]). We emphasize, hough, ha he class A1,
which is he smalles among he Apclasses, is he na u al one o conside when
a emp ing o p o e unweigh ed es ima es.
Theo em 1.3 can be gene alized o gi e “o -diagonal” esul s. In he classi-
cal se ing, he ex apola ion heo em o Rubio de F ancia was ex ended in his
manne by Ha bou e, Mac´ıas and Sego ia [24].
Theo em 1.8. Gi en a amily Fand an open se Ω⊂Rn, assume ha o
some p0and q0,0< p0≤q0<∞, and e e y weigh w∈A1,
(1.8) ZΩ
(x)q0w(x)dx1/q0
≤C0ZΩ
g(x)p0w(x)p0/q0dx1/p0
,( , g)∈F.
Gi en p(·)∈P0(Ω) such ha p0< p−≤p+< p0q0/(q0−p0), de ine he
unc ion q(·)by
(1.9) 1
p(x)−1
q(x)=1
p0
−1
q0
, x ∈Ω.
I q(x)/q00∈B(Ω), hen o all ( , g)∈Fsuch ha ∈Lq(·)(Ω),
(1.10) k kq(·),Ω≤Ckgkp(·),Ω.
Rema k 1.9. As be o e, (1.10) holds i p(·) sa is ies (1.1) and (1.2).
The boundedness o classical ope a o s on a iable Lpspaces 245
We can gene alize Theo em 1.3 by combining i wi h he wo ex apola ion
heo ems discussed abo e. This is possible since A1⊂Ap, 1 < p ≤ ∞. This has
wo ad an ages. Fi s , i makes clea ha he hypo heses which mus be sa is ied
co espond o hose o he known weigh ed no m inequali ies; see, in pa icula ,
he applica ions discussed in Sec ion 2 below. Second, as in [11], we a e able o
p o e ec o - alued inequali ies in a iable Lpspaces wi h essen ially no addi ional
wo k. All such inequali ies a e new.
Co olla y 1.10. Gi en a amily Fand an open se Ω⊂Rn, assume ha
o some p0,0< p0<∞, and o e e y w∈A∞,
(1.11) ZΩ
(x)p0w(x)dx ≤C0ZΩ
g(x)p0w(x)dx, ( , g)∈F.
Le p(·)∈P0(Ω) be such ha he e exis s 0< p1< p−wi h p(·)/p10∈B(Ω).
Then o all ( , g)∈Fsuch ha ∈Lp(·)(Ω),
(1.12) k kp(·),Ω≤Ckgkp(·),Ω.
Fu he mo e, o e e y 0<q<∞and sequence {( j, gj)}j⊂F,
(1.13) 


X
j
( j)q1/q


p(·),Ω
≤C


X
j
(gj)q1/q


p(·),Ω
.
Co olla y 1.11. Gi en a amily Fand an open se Ω⊂Rn, assume ha
(1.11) holds o some 1< p0<∞, o e e y w∈Ap0and o all ( , g)∈F. Le
p(·)∈P(Ω) be such ha he e exis s 1< p1< p−wi h p(·)/p10∈B(Ω).
Then (1.12) holds o all ( , g)∈Fsuch ha ∈Lp(·)(Ω). Fu he mo e, o
e e y 1<q<∞and {( j, gj)}j∈F, he ec o - alued inequali y (1.13) holds.
The es o his pape is o ganized as ollows. To illus a e he powe o ou
esul s, we i s conside some applica ions. In Sec ion 2 we gi e a numbe o
examples o ope a o s which a e bounded on Lp(·). These esul s a e immedia e
consequences o he abo e esul s and he heo y o weigh ed no m inequali ies.
Some o hese ha e been p o ed by o he s, bu mos a e new. We also p o e ec o -
alued inequali ies o hese ope a o s, all o which a e new esul s. In Sec ion 3 we
p esen an applica ion o pa ial di e en ial equa ions: we ex end he Calde ´on–
Zygmund inequali y (see [5], [22]) o solu ions o 4u= wi h ∈Lp(·)(Ω). In
Sec ion 4 we gi e an applica ion o he heo y o Sobole spaces: we show ha
he Calde ´on ex ension heo em (see [2], [4]) holds in a iable Sobole spaces. In
Sec ion 5 we p o e Theo ems 1.3 and 1.8. Ou p oo is adap ed om he a gumen s
gi en in [11]. Finally, in Sec ion 6 we p o e Co olla ies 1.10 and 1.11.
Th oughou his pape , we will make use o he basic p ope ies o a iable
Lpspaces, and will s a e some esul s as needed. Fo a de ailed discussion o hese
spaces, see Ko ´aˇcik and R´akosn´ık [30]. As we no ed abo e, in o de o emphasize
ha we a e dealing wi h a iable exponen s, we will always w i e p(·) ins ead o
p o deno e an exponen unc ion. Th oughou , Cwill deno e a posi i e cons an
whose exac alue may change a each appea ance.
246 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
2. Applica ions: Es ima es o classical ope a o s on Lp(·)
In his sec ion we gi e a numbe o applica ions o Theo ems 1.3 and 1.8,
and Co olla ies 1.10 and 1.11, o show ha a wide a ie y o classical ope a o s
a e bounded on he a iable Lpspaces. In he ollowing applica ions we will
impose di e en condi ions on he exponen s p(·) o gua an ee he co esponding
es ima es. In mos o he cases, i will su ice o assume ha p(·)∈B(Rn), o
in pa icula ha p(·) sa is ies (1.1) and (1.2).
As we no ed in he ema ks ollowing Theo em 1.3, o p o e hese applica ions
we will need o use densi y a gumen s. In doing so we will use he ollowing ac s:
(1) L∞
c, bounded unc ions o compac suppo , and C∞
c, smoo h unc ions o
compac suppo , a e dense in Lp(·)(Ω). See Ko ´aˇcik and R´akosn´ık [30].
(2) I p+<∞and ∈Lp+(Ω) ∩Lp−(Ω), hen ∈Lp(·)(Ω). This ollows om
he ac ha | (x)|p(x)≤ | (x)|p+χ{| (x)|≥1}+| (x)|p−χ{| (x)|<1}.
2.2. The Ha dy–Li lewood maximal unc ion. I is well known ha
o 1 <p<∞and o w∈Ap,
ZRn
M (x)pw(x)dx ≤CZRn
(x)pw(x)dx.
F om Co olla y 1.11 wi h he pai s (M , | |), we ge ec o - alued inequali ies
o Mon Lp(·), p o ided he e exis s 1 < p1< p−wi h p(·)/p10∈B(Rn);
by Theo em 1.2, his is equi alen o p(·)∈B(Rn). To apply Co olla y 1.11 we
need o es ic he pai s o unc ions ∈L∞
c, bu since hese o m a dense subse
we ge he desi ed es ima e o all ∈Lp(·)(Rn).
Co olla y 2.1. I p(·)∈B(Rn), hen o all 1< q < ∞,



X
j
(M j)q1/q


p(·),Rn
≤C


X
j
| j|q1/q


p(·),Rn
.
Rema k 2.2. F om Co olla y 1.11 we also ge one o he implica ions o
Theo em 1.2: i p(·)/p10∈B(Rn) hen p(·)∈B(Rn). I is e y emp ing o
specula e ha all o Theo em 1.2 can be p o ed ia ex apola ion, bu we ha e
been unable o do so.
2.2. The sha p maximal ope a o . Gi en a measu able unc ion and
a cube Q, de ine
Q=1
|Q|ZQ
(y)dy,
and he sha p maximal ope a o by
M# (x) = sup
x3Q
1
|Q|ZQ
| (y)− Q|dy.
The boundedness o classical ope a o s on a iable Lpspaces 247
The sha p maximal ope a o was in oduced by Fe e man and S ein [20], who
showed ha o all p, 0 <p<∞, and w∈A∞,
ZRn
M (x)pw(x)dx ≤CZRn
M# (x)pw(x)dx.
(Also see Jou n´e [26].) The e o e, by Co olla y 1.10 wi h he pai s (M , M# ),
∈L∞
c(Rn), and by Theo em 1.2 we ha e he ollowing esul .
Co olla y 2.3. Le p(·)∈P0(Rn)be such ha he e exis s 0< p1< p−
wi h p(·)/p1∈B(Rn). Then,
(2.1) kM kp(·),Rn≤CkM# kp(·),Rn,
and o all 0<q<∞,
(2.2) 


X
j
(M j)q1/q


p(·),Rn
≤C


X
j
(M# j)q1/q


p(·),Rn
.
Rema k 2.4. Co olla y 2.3 gene alizes esul s due o Diening and R˚uˇziˇcka
[15, Theo em 3.6] and Diening [14, Theo em 8.10], who p o ed (2.1) wi h M
eplaced by on he le -hand side and unde he assump ions ha p(·) and
p0(·)∈B(Rn) wi h 1 < p−≤p+<∞in he i s pape and p(·)∈B(Rn)
in he second. No ice ha ou esul is mo e gene al since we allow p(·) o go
below 1 and we only need p(·)/p10∈B(Rn) o some small alue 0 < p1< p−.
Fu he mo e, we au oma ically ob ain he ec o - alued inequali ies gi en in (2.2).
2.3. Singula in eg al ope a o s. Gi en a locally in eg able unc ion K
de ined on Rn {0}, suppose ha he Fou ie ans o m o Kis bounded, and K
sa is ies
(2.3) |K(x)| ≤ C
|x|n,|∇K(x)| ≤ C
|x|n+1 , x 6= 0.
Then he singula in eg al ope a o T, de ined by T (x) = K∗ (x), is a
bounded ope a o on weigh ed Lp. Mo e p ecisely, gi en 1 <p<∞, i w∈Ap,
hen
(2.4) ZRn
|T (x)|pw(x)dx ≤CZRn
| (x)|pw(x)dx.
(Fo de ails, see [17], [21].)
F om Co olla y 1.11, we ge ha Tis bounded on a iable Lpp o ided he e
exis s 1 < p1< p−wi h p(·)/p10∈B(Rn); by Theo em 1.2 his is equi alen
o p(·)∈B(Rn). Again, o apply he co olla y we need o es ic ou sel es o a
sui able dense amily o unc ions. We use he ac ha C∞
cis dense in Lp(·)(Rn),
and he ac ha i ∈C∞
c, hen T ∈T1<p<∞Lp⊂Lp(·)(Rn).
254 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
We begin wi h a ew de ini ions and a lemma. Gi en p(·)∈P(Ω) and a
na u al numbe k, de ine he a iable Sobole space Wk,p(·)(Ω) o be he se o
all unc ions ∈Lp(·)(Ω) such ha
X
|α|≤k
kDα kp(·),Ω<+∞,
whe e he de i a i es a e unde s ood in he sense o dis ibu ions.
Gi en a unc ion which is wice di e en iable (in he weak sense), we de ine
o i= 1,2,
Di =X
|α|=i
(Dα )21/2
.
We need he ollowing auxilia y esul whose p oo can be ound in [30].
Lemma 3.1. I Ω⊂Rnis a bounded domain, and i p(·), q(·)∈P(Ω) a e
such ha p(x)≤q(x),x∈Ω, hen k kp(·),Ω≤(1 + |Ω|)k kq(·),Ω.
Theo em 3.2. Gi en an open se Ω⊂Rn,n≥3, suppose p(·)∈P(Ω)
wi h p+< n/2sa is ies (1.1) and (1.2). I ∈Lp(·)(Ω), hen he e exis s a
unc ion u∈Lq(·)(Ω), whe e
(3.1) 1
p(x)−1
q(x)=2
n,
such ha
(3.2) 4u(x) = (x),a.e. x∈Ω.
Fu he mo e,
kD2ukp(·),Ω≤Ck kp(·),Ω,(3.3)
kD1uk (·),Ω≤Ck kp(·),Ω,(3.4)
kukq(·),Ω≤Ck kp(·),Ω,(3.5)
whe e 1
p(x)−1
(x)=1
n.
In pa icula , i Ωis bounded, hen u∈W2,p(·)(Ω).
P oo . Ou p oo oughly ollows he p oo in he se ing o Lebesgue spaces
gi en by Gilba g and T udinge [22], bu also uses his esul in key s eps.
Fix ∈Lp(·)(Ω); wi hou loss o gene ali y we may assume ha k kp(·),Ω= 1.
Decompose as
= 1+ 2= χ{x:| (x)|>1}+ χ{x:| (x)|≤1}.

The boundedness o classical ope a o s on a iable Lpspaces 255
No e ha | i(x)| ≤ | (x)|and so k ikp(·),Ω≤1. Fu he , we ha e ha 1∈
Lp−(Ω) and 2∈Lp+(Ω) since, by he de ini ion o he no m in Lp(·)(Ω) and
since k kp(·),Ω= 1,
ZΩ
1(x)p−dx =Z{x∈Ω:| (x)|>1}
| (x)|p−dx ≤ZΩ
| (x)|p(x)dx ≤1,
ZΩ
2(x)p+dx =Z{x∈Ω:| (x)|≤1}
| (x)|p+dx ≤ZΩ
| (x)|p(x)dx ≤1.
Thus, we can sol e Poisson’s equa ion wi h 1and 2(see [22]): mo e p ecisely,
de ine
u1(x) = (Γ ∗ 1)(x), u2= (Γ ∗ 2)(x),
whe e Γ is he New onian po en ial,
Γ(x) = 1
n(2 −n)ωn
|x|2−n,
and ωnis he olume o he uni ball in Rn. Since p−and q−also sa is y (3.1),
by he Calde ´on–Zygmund inequali y on classical Lebesgue spaces, u1∈Lq−(Ω).
Simila ly, since p+and q+sa is y (3.1), u2∈Lq+(Ω). Le u=u1+u2; hen
u∈Lq−(Ω) + Lq+(Ω). Since u1and u2a e solu ions o Poisson’s equa ion,
4u(x) = 4u1(x) + 4u2(x) = 1(x) + 2(x) = (x),a.e. x∈Ω.
We show ha u∈Lq(·)(Ω) and ha (3.5) holds: by inequali y (2.12),
kukq(·),Ω≤ ku1kq(·),Ω+ku2kq(·),Ω
=1
n(2 −n)ωnkI2 1kq(·),Ω+kI2 2kq(·),Ω
≤Ck 1kp(·),Ω+k 2kp(·),Ω
≤C=Ck kp(·),Ω;
he las equali y holds since k kp(·),Ω= 1.
Simila ly, a di ec compu a ion shows ha o any mul i-index α,|α|= 1,
|DαΓ(x)| ≤ 1
n ωn
|x|1−n.
The e o e,
|Dαu(x)| ≤ |Dα(Γ ∗ 1)(x)|+|Dα(Γ ∗ 2)(x)|
=|(DαΓ∗ 1)(x)|+|(DαΓ∗ 2)(x)|
≤1
n ωnI1(| 1|)(x) + I1(| 2|)(x).
256 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
So again by inequali y (2.12) we ge
kDαuk (·),Ω≤Ck 1kp(·),Ω+k 2kp(·),Ω≤C,
which yields inequali y (3.4).
Gi en a mul i-index α,|α|= 2, ano he compu a ion shows ha DαΓ is a
singula con olu ion ke nel which sa is ies (2.3). The e o e, he ope a o
Tαg(x) = (DαΓ∗g)(x) = Dα(Γ ∗g)(x)
is singula in eg al ope a o , and as be o e (3.3) ollows om inequali y (2.5) and
Rema k 2.6 applied o 1and 2.
Finally, i Ω is bounded, since p(x)≤q(x) and p(x)≤ (x), x∈Ω, by
Lemma 3.1 we ha e ha u∈W2,p(·)(Ω).
Rema k 3.3. In he p e ious es ima es we could ha e wo ked di ec ly wi h .
Had we done so, howe e , we would ha e had o check ha all he in eg als
appea ing we e absolu ely con e gen . The ad an age o decomposing as 1+ 2
is ha we did no need o pay a en ion o his since 1∈Lp−(Ω), 2∈Lp+(Ω).
We also wan o s ess ha u1and u2, as solu ions o Poisson’s equa ion wi h
1∈Lp−(Ω) and 2∈Lp+(Ω), sa is y Lebesgue space es ima es. Fo ins ance,
as no ed abo e, u∈Lq−(Ω) + Lq+(Ω). Howe e , we ha e ac ually p o ed mo e,
since Lq(·)(Ω) is a smalle space. Simila ema ks hold o he i s and second
de i a i es o u.
4. The Calde ´on ex ension heo em
In his sec ion we s a e and p o e he Calde ´on ex ension heo em o a iable
Sobole spaces. Ou p oo ollows closely he p oo o he esul in he classical se -
ing; see, o example, R. Adams [2] o Calde ´on [4]. Fi s , we gi e wo de ini ions
and a lemma.
De ini ion 4.1. Gi en a poin x∈Rn, a ini e cone wi h e ex a x,Cx,
is a se o he o m
Cx=B1∩ {x+λ(y−x) : y∈B2, λ > 0},
whe e B1is an open ball cen e ed a x, and B2is an open ball which does no
con ain x.
De ini ion 4.2. An open se Ω ⊂Rnhas he uni o m cone p ope y i
he e exis s a ini e collec ion o open se s {Uj}(no necessa ily bounded) and an
associa ed collec ion {Cj}o ini e cones such ha he ollowing hold:
(1) he e exis s δ > 0 such ha
Ωδ={x∈Ω : dis (x, ∂Ω) < δ} ⊂ S
j
Uj;
(2) o e e y index jand e e y x∈Ω∩Uj,x+Cj⊂Ω.
The boundedness o classical ope a o s on a iable Lpspaces 257
An example o a se Ω wi h he uni o m cone p ope y is any bounded se
whose bounda y is locally Lipschi z. (See Adams [2].)
Finally, in gi ing ex ension heo ems o a iable Lpspaces, we mus wo y
abou ex ending he exponen unc ion p(·). The ollowing esul shows ha his
is always possible, p o ided ha p(·) sa is ies (1.1) and (1.2).
Lemma 4.3. Gi en an open se Ω⊂Rnand p(·)∈P(Ω) such ha (1.1)
and (1.2) hold, he e exis s a unc ion ˜p(·)∈P(Rn)such ha :
(1) ˜psa is ies (1.1) and (1.2);
(2) ˜p(x) = p(x),x∈Ω;
(3) ˜p−=p−and ˜p+=p+.
Rema k 4.4. Diening [13] p o ed an ex ension heo em o exponen s p(·)
which sa is y (1.1), p o ided ha Ω is bounded and has Lipschi z bounda y. I
would be in e es ing o de e mine i e e y exponen p(·)∈B(Ω) can be ex ended
o an exponen unc ion in B(Rn).
P oo . Since p(·) is bounded and uni o mly con inuous, by a well-known e-
sul i ex ends o a con inuous unc ion on Ω. S aigh o wa d limi ing a gumen s
show ha his ex ension sa is ies (1), (2) and (3).
The ex ension o p(·) on Ω o ˜p(·) de ined on all o Rn ollows om a
cons uc ion due o Whi ney [45] and desc ibed in de ail in S ein [42, Chap e 6].
Fo ease o e e ence, we will ollow S ein’s no a ion. We i s conside he case
when Ω is unbounded; he case when Ω is bounded is simple and will be ske ched
below.
When Ω is unbounded, (1.2) is equi alen o he exis ence o a cons an p∞,
p−≤p∞≤p+, such ha o all x∈Ω,
|p(x)−p∞| ≤ C
log(e+|x|).
De ine a new unc ion (·) by (x) = p(x)−p∞. Then (·) is s ill bounded
( hough no longe necessa ily posi i e), s ill sa is ies (1.1) on Ω and sa is ies
(4.1) | (x)| ≤ C
log(e+|x|).
We will ex end o all o Rn. I we de ine ω( ) = 1/log(e/2 ), 0 < ≤1/2,
and ω( ) = 1 o ≥1/2, hen a s aigh o wa d calcula ion shows ha ω( )/
is a dec easing unc ion and ω(2 )≤C ω( ). Fu he , since log(e/2 )≈log(1/ ),
0< < 1/2, and since is bounded, | (x)− (y)| ≤ Cω(|x−y|) o all x, y ∈Ω.
The e o e, by Co olla y 2.2.3 in S ein [42, p. 175], he e exis s a unc ion ˜ (·)
on Rnsuch ha ˜ (x) = (x), x∈Ω, and such ha ˜ (·) sa is ies (1.1). Fo
x∈Rn Ω, ˜ (x) is de ined by he sum
˜ (x) = X
k
(pk)ϕ∗
k(x),
258 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
whe e {Qk}a e he cubes o he Whi ney decomposi ion o Rn Ω, {ϕ∗
k}is he
pa i ion o uni y subo dina e o his decomposi ion, and each poin pk∈Ω is
such ha dis (pk, Qk) = dis (Ω, Qk).
I ollows immedia ely om his de ini ion ha o all x∈Rn, −≤˜ (x)≤
+. Howe e , ˜ (·) need no sa is y (4.1) so we mus modi y i sligh ly. To do so we
need he ollowing obse a ion: i 1, 2a e unc ions such ha | i(x)− i(y)| ≤
Cω(|x−y|), x, y ∈Rn,i= 1,2, hen min( 1, 2) and max( 1, 2) sa is y he same
inequali y. The p oo o his obse a ion consis s o a numbe o e y simila cases.
Fo ins ance, suppose min 1(x), 2(x)= 1(x) and min 1(y), 2(y)= 2(y).
Then
1(x)− 2(y)≤ 2(x)− 2(y)≤Cω(|x−y|),
2(y)− 1(x)≤ 1(y)− 1(x)≤Cω(|x−y|).
Hence,
min 1(x), 2(x)−min 1(y), 2(y)=| 1(x)− 2(y)| ≤ Cω(|x−y|).
I ollows immedia ely om his obse a ion ha
s(x) = maxmin(˜ (x), C/ log(e+|x|),−C/ log(e+|x|)
sa is ies (1.1) and (4.1). The e o e, i we de ine
˜p(x) = s(x) + p∞,
hen (1), (2) and (3) hold.
Finally, i Ω is bounded, we de ine (x) = p(x)−p+and epea he abo e
a gumen essen ially wi hou change.
Theo em 4.5. Gi en an open se Ω⊂Rnwhich has he uni o m cone
p ope y, and gi en p(·)∈P(Ω) such ha (1.1) and (1.2) hold, hen o any
na u al numbe k he e exis s an ex ension ope a o
Ek:Wk,p(·)(Ω) →Wk,p(·)(Rn),
such ha Eku(x) = u(x), a.e. x∈Ω, and
kEkukp(·),Rn≤C(p(·),k,Ω)kukp(·),Ω.
The p oo o Theo em 4.5 in a iable Sobole spaces is nea ly iden ical o ha
in he classical se ing. (See Adams [2].) The p oo , beyond calcula ions, equi es
he ollowing ac s which ou hypo heses insu e a e ue.
The boundedness o classical ope a o s on a iable Lpspaces 259
– By Lemma 4.3, p(·) immedia ely ex ends o an exponen unc ion on Rn.
– Func ions in C∞(Ω) a e dense in Wk,p(·)(Ω). By ou hypo heses, he max-
imal ope a o is bounded on Lp(·)(Ω), and he densi y o C∞(Ω) ollows
om his by he s anda d a gumen (c . Zieme [47]). Fo mo e de ails, see
Diening [12] o C uz-U ibe and Fio enza [9].
– I ϕis a smoo h unc ion on Rn {0}wi h compac suppo , and i he e
exis s ε > 0 such ha on Bε(0), ϕis a homogeneous unc ion o deg ee k,
k > −n, hen kϕ∗ kp(·),Ω≤Cp(·), ϕk kp(·),Ω. This again ollows om
he ac ha he maximal ope a o is bounded on Lp(·)(Ω), and om he well-
known inequali y |ϕ∗ (x)| ≤ CM (x). Fo mo e de ails, see C uz-U ibe and
Fio enza [9].
– Singula in eg al ope a o s wi h ke nels o he o m
K(x) = G(x)
|x|n,
whe e Gis bounded on Rn {0}, has compac suppo , is homogeneous o
deg ee ze o on BR(0) {0} o some R > 0, and has RSRG dx = 0, a e
bounded on Lp(·)(Ω). Such ke nels a e essen ially he same as hose gi en by
(2.9), and as discussed abo e, ou hypo heses imply ha hey a e bounded.
Rema k 4.6. I p(·) sa is ies (1.1), hen C∞
c(Rn) is dense in Wk,p(·)(Rn).
(See [9], [41].) Hence, i he hypo heses o Theo em 4.5 hold, hen i ollows
immedia ely ha he se {uχΩ:u∈C∞
c(Rn)}is dense in Wk,p(·)(Ω). Howe e
his esul is ue unde much weake hypo heses; see [9], [18], [19], [25], [46] o
de ails.
5. P oo o Theo ems 1.3 and 1.8
Since Theo em 1.3 is a pa icula case o Theo em 1.8 wi h p0=q0, i su ices
o p o e he second esul .
We need wo ac s abou a iable Lpspaces. Fi s , i p(·), q(·)∈P0(Ω)
and p(x)/q(x) = , hen i ollows om he de ini ion o he no m ha
(5.1) k k
p(·),Ω=
| | 
q(·),Ω.
Second, gi en p(·)∈P(Ω), we ha e he gene alized H¨olde ’s inequali y
(5.2) ZΩ
| (x)g(x)|dx ≤1 + 1
p−
−1
p+k kp(·),Ωkgkp0(·),Ω,
and he “duali y” ela ionship
(5.3) k kp(·),Ω≤sup
gZΩ
(x)g(x)dx≤1 + 1
p−
−1
p+k kp(·),Ω,

260 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
whe e he sup emum is aken o e all g∈Lp0(·)(Ω) such ha kgkp0(·),Ω= 1. Fo
p oo s o hese esul s, see Ko ´aˇcik and R´akosn´ık [30].
The p oo o Theo em 1.8 begins wi h a e sion o a cons uc ion due o Rubio
de F ancia [38] (also see [11], [21]). Fix p(·)∈P0(Ω) such ha p−> p0, and le
¯p(x) = p(x)/p0. De ine q(·) as in (1.9), and le ¯q(x) = q(x)/q0. By assump ion,
he maximal ope a o is bounded on L¯q0(·)(Ω), so he e exis s a posi i e cons an
Bsuch ha
kM k¯q0(·),Ω≤Bk k¯q0(·),Ω.
De ine a new ope a o Ron L¯q0(·)(Ω) by
Rh(x) =
∞
X
k=0
Mkh(x)
2kBk,
whe e, o k≥1, Mk=M◦M◦· · ·◦Mdeno es ki e a ions o he maximal ope -
a o , and M0is he iden i y ope a o . I ollows immedia ely om his de ini ion
ha :
(a) i his non-nega i e, h(x)≤Rh(x);
(b) kRhk¯q0(·),Ω≤2khk¯q0(·),Ω;
(c) o e e y x∈Ω, M(Rh)(x)≤2BRh(x), so Rh∈A1wi h an A1cons an
ha does no depend on h.
We can now a gue as ollows: by (5.1) and (5.3),
k kq0
q(·),Ω=k q0k¯q(·),Ω≤sup ZΩ
(x)q0h(x)dx,
whe e he sup emum is aken o e all non-nega i e h∈L¯q0(·)(Ω) wi h khk¯q0(·),Ω=
1. Fix any such unc ion h; i will su ice o show ha
ZΩ
(x)q0h(x)dx ≤Ckgkq0
p(·),Ω
wi h he cons an Cindependen o h. Fi s no e ha by (a) abo e we ha e ha
(5.4) ZΩ
(x)q0h(x)dx ≤ZΩ
(x)q0Rh(x)dx.
By (5.2), (b), and since ∈Lq(·)(Ω),
ZΩ
(x)q0Rh(x)dx ≤Ck q0k¯q(·),ΩkRhk¯q0(·),Ω
≤Ck kq0
q(·),Ωkhk¯q0(·),Ω
≤Ck kq0
q(·),Ω<∞.
The boundedness o classical ope a o s on a iable Lpspaces 261
The e o e, we can apply (1.8) o he igh -hand side o (5.4) and again apply (5.2),
his ime wi h exponen ¯p(·):
ZΩ
(x)q0Rh(x)dx ≤CZΩ
g(x)p0Rh(x)p0/q0dxq0/p0
≤Ckgp0kq0/p0
¯p(·),Ωk(Rh)p0/q0kq0/p0
¯p0(·),Ω
=Ckgkq0
p(·),Ωk(Rh)p0/q0kq0/p0
¯p0(·),Ω.
To comple e he p oo , we need o show ha k(Rh)p0/q0kq0/p0
¯p0(·),Ωis bounded by a
cons an independen o h. Bu i ollows om (1.9) ha o all x∈Ω,
¯p0(x) = p(x)
p(x)−p0
=q0
p0
q(x)
q(x)−q0
=q0
p0
¯q0(x).
The e o e,
k(Rh)p0/q0kq0/p0
¯p0(·),Ω=kRhk¯q0(·),Ω≤Ckhk¯q0(·),Ω=C.
This comple es ou p oo .
6. P oo o Co olla ies 1.10 and 1.11
The p oo s o Co olla ies 1.10 and 1.11 equi e he mo e gene al e sions o
he ex apola ion heo ems discussed in he in oduc ion. Fo he con enience o
he eade we s a e hem bo h he e.
Theo em 6.1. Gi en a amily Fand an open se Ω⊂Rn, assume ha o
some p0,0< p0<∞, and o e e y w∈A∞,
(6.1) ZΩ
(x)p0w(x)dx ≤C0ZΩ
g(x)p0w(x)dx, ( , g)∈F.
Then o all 0<p<∞and w∈A∞,
(6.2) ZΩ
(x)pw(x)dx ≤C0ZΩ
g(x)pw(x)dx, ( , g)∈F.
Fu he mo e, o e e y 0< p, q < ∞,w∈A∞, and sequence {( j, gj)}j⊂F,
(6.3) 


X
j
( j)q1/q


Lp(w,Ω)
≤C


X
j
(gj)q1/q


Lp(w,Ω)
.
Theo em 6.2. Gi en a amily Fand an open se Ω⊂Rn, assume ha
o some p0,1< p0<∞, and o e e y w∈Ap0,(6.1) holds. Then o e e y
1< p < ∞and w∈Ap,(6.2) holds. Fu he mo e, o e e y 1< p, q < ∞,
w∈Ap, and sequence {( j, gj)}j⊂F,(6.3) holds.
262 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez
Theo em 6.1 is p o ed in [11]. The o iginal s a emen o Theo em 6.2 is
only o pai s o he o m (|T |, ), and does no include he ec o - alued es i-
ma e (6.3). (See [17], [21], [38].) Howe e , an examina ion o he p oo s shows
ha hey hold wi hou change when applied o pai s ( , g)∈F. Fu he mo e, as
we no ed be o e, his app oach immedia ely yields he ec o - alued inequali ies:
gi en a amily Fand 1 <q<∞, de ine he new amily Fq o consis o he
pai s (Fq, Gq), whe e
Fq(x) = X
j
( j)q1/q
, Gq(x) = X
j
(gj)q1/q
,{( j, gj)}j⊂F.
Clea ly, inequali y (6.1) holds o Fqwhen p0=q, so by ex apola ion we
ge (6.3).
Co olla y 1.10 ollows immedia ely om Theo ems 1.3 and 6.1. Since (1.11)
holds o some p0, by Theo em 6.1 i holds o all 0 <p<∞and o all w∈A∞.
The e o e, we can apply Theo em 1.3 wi h p1in place o p0 o ob ain (1.12).
To p o e he ec o - alued inequali y (1.13), no e ha by (6.3) we can apply
Theo em 1.3 o he amily Fqde ined abo e, again wi h p1in place o p0.
In exac ly he same way, Co olla y 1.11 ollows om Theo ems 1.3 and 6.2.
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